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I've got my new Totaline Wireless stat installed and working on my gas furnace. My old stat, a Honeywell, only had 3 wires going to it (W,G,R). When I first hooked up the new stat and ran the fan I had the problem with the fan cycling on and off every few seconds. The lights on the new stat would also dim. If I tried to turn the heat on the furnace would turn off and the stat would lose power. From the manual and what I read online it sounded like a 270 ohm 10 watt resistor between 2 of the terminals on the furnace would fix it. It did not. I removed the resistor and hooked up the, previously unused, C terminal wire to the new stat. This fixed the problem and it works fine now. I'm am curious if this was the correct thing to do and why this fixed the problem. Any thoughts?

The common wire supplies power to the t-stat - very common on models that have LED indicators.

The batteries are there to keep the time and program in the event of a power failure. When there is a call for heating/cooling, additional energy is required to keep the relay closed. (That "click" you hear when the system kicks in is a small relay closing)

Many thermostats; typically the cheaper ones sold at home improvement stores are either battery powered or power stealing. (They "steal" power from the control circuits)

I can only assume that you dont have air conditioning  if not, Y should also be hooked up.

why? because solid state relays alow a trickle of voltages through, the low current through the relay will cause problems on a board that only uses a few miliamps to operate, the resister acts as a load and eliminates the voltage.

Resistors can also be used to create a voltage divider network. They can provide specified voltages and currents with only one voltage source. There are different types of resistors; variable, tapped wire wound, fixed wire wound, thin film technique, and fixed composition which is very popular. Resistors are color coded. The first band will determine the first significant figure, the second will determine the next significant figure, the third band will determine the multiplier value of the first two significant figures, the fourth band determines tolerance of the resistor, the fifth determines the reliability level(only for military specs). The power rating is usually determined by the physical size of the resistor. Ohm's law E = I x R.